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Between Filamentation and Two-Stream instabilities in e-beam/plasma interaction

Between Filamentation and Two-Stream instabilities in e-beam/plasma interaction. A. Bret, M.-C. Firpo and C. Deutsch Laboratoire de Physique des Gaz et des Plasmas – Paris Orsay Université. Filamentation, Weibel. ?. k. Two-Stream. The problem of k -orientation. Beams. Formalism.

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Between Filamentation and Two-Stream instabilities in e-beam/plasma interaction

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  1. Between Filamentation and Two-Stream instabilities in e-beam/plasma interaction A. Bret, M.-C. Firpo and C. Deutsch Laboratoire de Physique des Gaz et des Plasmas – Paris Orsay Université

  2. Filamentation, Weibel ? k Two-Stream The problem of k-orientation Beams

  3. Formalism • Linear Maxwell-Relativistic Vlasov system in 3D • Collisionless – Fixed plasma ions • Analytical • Dielectic tensor e (k,w) non-diagonal • Det T(e, k) = 0 : proper frequencies • E along eigenvectors : gives angle (k,E)

  4. Filamentation k ? k E Two-Stream The (k,E) angle

  5. Dispersion Relation Det T(e, k) = 0  (w2eyy - k2c2)× [(w2exx - kz2c2)(w2ezz - kx2c2) - (w2exz + kxkzc2)2] = 0

  6. Two Branches Branch 1 (w2eyy - k2c2) = 0 Branch 2 [(w2exx - kz2c2)(w2ezz - kx2c2) - (w2exz + kxkzc2)2] = 0

  7. Model 1: T = 0 (Fluid) Beam : nbd(vx)d(vy)d(vz-Vb) Plasma : npd(vx)d(vy)d(vz+Vp) nbVb = npVp

  8. Model 2: Hot Plasma Beam : nbd(vx)d(vy)d(vz-Vb) Plasma : npd(vz+Vp) × [Q(vx+Vt)+ Q(vx-Vt)] × [Q(vy+Vt)+ Q(vy-Vt)] nbVb = npVp 2Vt

  9. Model 3: Hot Plasma + Relativistic Beam Beam : Relativistic energyE ~ 2 MeV (gb ~ 4) Plasma : Non relativisticT ~ 2 keV

  10. k E // y Beam UNSTABLE Branch 1 behavior (w2eyy - k2c2) = 0 • Stable at T = 0 • Unstable T ≠ 0 for small angles • No relativistic effects (model 3 = model 2) • Purely transverse mode in ANY model • Original « Weibel » instability

  11. qk k Weibel result Branch 1 behavior Z = kVb /wp Beam 103d d dRm Vtp/Vb r = Vtp/Vb, b=Vb/c

  12. Branch 2 behavior • Yields unstable modes for all models and all angles • Bridge between Two-Stream and Filamentation [(w2exx - kz2c2)(... x k E Beam z

  13. Between two-streamand filamentation, T=0 d/wp Beam Z = kVb /wp

  14. Between two-streamand filamentation, T=0 d/wp Beam Z = kVb /wp

  15. Between Two-Streamand filamentation, T≠0 d/wp Beam

  16. Between Two-Streamand filamentation, T≠0 d/wp Beam

  17. Tanf = 1+nb /np Vth /Vb fp/2 Vth0 Critical angle f Angle comes when 2 dispersion function singularities cross High Z instability just is just shifted in another direction

  18. Between two-stream and filamentation, T≠0Relativistic effects d Beam

  19. Same Angle Between two-stream and filamentation, T≠0Relativistic effects d Beam

  20. Zxm = Vb/ c (Vth /Vb)1/2 Zxm 1 Highest Growth Rate f BEAM

  21. Longitudinal approximation OK below f (k,E) angle f BEAM

  22. df a1/3 /gb dp/2  (a/gb)1/2 dm d0 dp/2= df  (a/gb)1/3 Growth Rates Scaling d a = nb/np b = Vb/c gb=(1-b2)-1/2

  23. k Which effect ? Important effect k No effect Transverse Beam Temp. Effects Vtb Beam

  24. Beam Temp. Effects 1+nb /np Tanf = (Vthb/gb+Vth)/Vb Beam Temp. damps growth rate beyond f

  25. Beam Temp. damps instabilities beyond f. Longitudinal approximation fails beyond f. Longitudinal approximation even better with hot beam. Non relativistic beam Vtb=0 Vtb=Vb/30 Vtb=Vb/30 + k // E

  26. Conclusion • Electromagnetic formalism • Exhaustive instabilities search • Weibel Branch • TSF Branch • TSF Branch: Two k-oblique effects • Critical angle f • Max. Growth Rate for oblique k (Vb~c) • Longi. Approx. Fails beyond f • Beam Temp. damps instabilities beyond f, not bellow • Maxwelian distribution, Collisions, Density gradient

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